1Solving Right TrianglesDateTopicApplications of Trigonometric Functions(4.8)Solve the right triangle shown.Find the measure of angle B. B 90º A B 90º 34.5º B 55.5º We have a known angle, an unknown opposite side, and a known adjacent side, use ? the tangent function. tan 34.5º a/10.5 10.5 tan 34.5 º a 7.22 a 2Text Example cont.We need to find c.We have a known angle, a known adjacent side, and an unknown hypotenuse, we use ? the cosine function. cos 34.5 º 10.5/c c 10.5/cos 34.5 º c 12.74 So, B 55.5º, a 7.22, and c 12.74 3Text ExampleNUse the figure to find a. the bearing from O to B. b. the bearing from O to A.40ºBA20ºa.WEOWe need the angle between the ray OB and the north-south line through O. The measurement of this angle is given to be 40º. This angle is west of that line. Thus, the bearing from O to B is N 40º W75ºDC25ºSWe need the angle between the ray OA and the north-south line through O. This angle measures 90º 20º, or 70º. This angle is east of the north-south line. Thus the bearing from O to A is N 70º Eb. 4Text ExampleNUse the figure to find c. the bearing from O to C. d. the bearing from O to D.40ºBA20ºc.WEOWe need the angle between the ray OC and the north-south line through O. This angle measures 90º 75º, or 15º and is west of the north-south line. Thus, the bearing from O to C is S 15º W75ºDC25ºSWe need the angle between the ray OD and the north-south line through O. This angle measures 25º. This angle is east of the north-south line. Thus the bearing from O to A is S 25º Ed. 5Complete Student CheckpointYou leave the entrance to a system of hiking trails and hike 2.3 miles on a bearing of S 31º W. Then the trails turns 90º clockwise and you hike 3.5 miles on a bearing of N 59º W. At that time a. How far are you from the entrance to the trail system?Use the pythagorean theorem to find the distance. 6Complete Student Checkpointb. What is your bearing from the entrance to the trail system?Your bearing from the entrance to the trail system is 7Simple Harmonic Motion

An object that moves on a coordinate axis is in simple harmonic motion if its distance from the origin, d, at time t is given by either

d a cos ? t or d a sin ? t

The motion has amplitude a, the maximum displacement of the object from its rest position. The period of the motion is 2?/? , where ? gt 0. The period gives the time it takes for the motion to go through one complete cycle. It has frequency f given by f ?/2?, ?gt0. Equivalently, f 1/period

8Example

An object in simple harmonic motion has a frequency of 1/4 oscillation per minute and an amplitude of 8 ft. Write an equation in the form for the objects simple harmonic motion.

a 8 and the period is 4 minutes since it travels 1/4 oscillation per minute 9Example

A mass moves in simple harmonic motion described by the following equation, with t measured in seconds and d in centimeters. Find the maximum displacement, the frequency, and the time required for one cycle.

The frequency is1/6 cm per second.The maximum displacement is 8 cm, since a 8 The time required for one cycle is6 seconds. 10Complete Student CheckpointA ball on a spring is pulled six inches below its rest position and then released. The period for the motion is 4 seconds. Write the equation for the balls simple harmonic motion.a -6 11Applications of Trigonometric Functions

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